#7766618

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(7,10)(8,12);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,19)(15,20,25,28)(16,21,26,24)(17,22,27,23);(5,7)(6,8)(9,10)(11,12)(14,19)(17,27)(20,28)(21,24);(7,10)(8,12)(13,15)(14,20)(16,22)(17,24)(18,25)(19,28)(21,27)(23,26);(5,8,9,12)(6,10,11,7)(13,16)(14,21)(15,23)(17,20)(18,26)(19,24)(22,25)(27,28);(5,6)(7,12)(8,10)(9,11)(13,17)(14,22)(15,24)(16,20)(18,27)(19,23)(21,25)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7673814);